login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = 10*n^2 - 6*n + 1.
4

%I #41 Nov 09 2024 10:03:46

%S 5,29,73,137,221,325,449,593,757,941,1145,1369,1613,1877,2161,2465,

%T 2789,3133,3497,3881,4285,4709,5153,5617,6101,6605,7129,7673,8237,

%U 8821,9425,10049,10693,11357,12041,12745,13469,14213,14977,15761,16565,17389,18233,19097

%N a(n) = 10*n^2 - 6*n + 1.

%C Sequence found by reading the line from 5, in the direction 5, 29, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Jul 18 2012

%H Shawn A. Broyles, <a href="/A087348/b087348.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n)^2 = A033579(n)^2 + A033567(n)^2 = (4*A000326(n))^2 + (A033579(n) + A056220(n-1))^2.

%F From _Colin Barker_, Jun 30 2012: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f.: x*(5 + 14*x + x^2)/(1-x)^3. (End)

%F a(n) = 1 + A153784(n). - _Omar E. Pol_, Jul 18 2012

%F E.g.f.: exp(x)*(10*x^2 + 4*x + 1) - 1. - _Elmo R. Oliveira_, Oct 31 2024

%e a(3)=73 since 73^2 = 48^2 + 55^2 = (4*12)^2 + (48 + 7)^2. See 1st formula.

%t Table[10*n^2 - 6*n + 1, {n, 50}] (* _Paolo Xausa_, Jul 18 2024 *)

%o (PARI) a(n)=10*n^2-6*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000326, A033567, A033579, A056220, A085787, A153784.

%K nonn,easy

%O 1,1

%A _Charlie Marion_, Oct 20 2003

%E More terms from _Ray Chandler_, Oct 22 2003